Neutron star deformation due to poloidal-toroidal magnetic fields of arbitrary multipole order: a new analytic approach

A recipe is presented to construct an analytic, self-consistent model of a non-barotropic neutron star with a poloidal-toroidal field of arbitrary multipole order, whose toroidal component is confined in a torus around the neutral curve inside the star, as in numerical simulations of twisted tori. The recipe takes advantage of magnetic field aligned coordinates to ensure continuity of the mass density at the surface of the torus. The density perturbation and ellipticity of such a star are calculated in general and for the special case of a mixed dipole-quadrupole field as a worked example. The calculation generalizes previous work restricted to dipolar, poloidal-toroidal and multipolar, poloidal-only configurations. The results are applied, as an example, to magnetars whose observations (e.g. spectral features and pulse modulation) indicate that the internal magnetic fields may be at least one order of magnitude stronger than the external fields, as inferred from their spin-downs, and are not purely dipolar.